1232
J. Chem. Theory Comput. 2005, 1, 1232-1239
Modeling Proton Transfer in Zeolites: Convergence
Behavior of Embedded and Constrained Cluster
Calculations
Justin T. Fermann,† Teresa Moniz,† Oliver Kiowski,† Timothy J. McIntire,†
Scott M. Auerbach,*,†,‡ Thom Vreven,§ and Michael J. Frisch§
Departments of Chemistry and Chemical Engineering, UniVersity of Massachusetts,
Amherst, Massachusetts 01003, and Gaussian, Inc., 340 Quinnipiac Street,
Building 40, Wallingford, Connecticut 06492
Received May 2, 2005
Abstract: We have studied the convergence properties of embedded and constrained cluster
models of proton transfer in zeolites. We applied density functional theory to describe clusters
and ONIOM to perform the embedding. We focused on converging the reaction energy and
barrier of the O(1) to O(4) jump in H-Y zeolite as well as vibrational and structural aspects of
this jump. We found that using successively larger clusters in vacuo gives convergence of this
reaction energy to 14 ( 2 kJ mol-1 and the barrier to 135 ( 5 kJ mol-1 at a cluster size of 5 Å,
which contains 11 tetrahedral (Si or Al) atoms. We embedded quantum clusters of various sizes
in larger clusters with total radii in the range 7-20 Å, using the universal force field as the lower
level of theory in ONIOM. We found convergence to the same values as the constrained clusters,
without the use of reactive force fields or periodic boundary conditions in the embedding
procedure. For the reaction energy, embedded cluster calculations required smaller clusters
than in vacuo calculations, reaching converged reaction energies for quantum systems containing
at least 8 tetrahedral atoms. In addition, optimizations on embedded clusters required many
fewer cycles, and hence much less CPU time, than did optimizations on comparable constrained
clusters.
I. Introduction
Zeolites offer a versatile class of shape-selective catalysts
for important chemical processes such as petroleum cracking
and reforming.1,2 Many catalytic processes in zeolites are
activated by proton transfer from Brønsted acid sites: SiOH-Al. Progress in steering such reactions would be fueled
by enhanced understanding of the mechanisms that control
proton transfer in zeolites, which can be provided by
molecular modeling.3,4 Studying proton transfer in bare
zeolites is important for several reasons: (i) trends in catalytic
* Corresponding author e-mail: [email protected].
† Department of Chemistry, University of Massachusetts.
‡ Department of Chemical Engineering, University of Massachusetts.
§ Gaussian, Inc.
10.1021/ct0501203 CCC: $30.25
activity have been correlated with proton-transfer rates,5,6 (ii)
several intriguing discrepancies remain among experimental
probes of proton transfer in bare zeolites,7-10 and (iii)
computational methods needed to model complex reactions
in zeolites can be validated on this relatively simpler
process.11-14 An important issue in modeling zeolite electronic structures is how to represent the extended nature of
zeolites with tractable calculations. In this article, we
benchmark two approaches for computing the reaction energy
and barrier for the O(1) to O(4) proton transfer in H-Y
zeolite.
Quantum calculations on small clusters have long been
the staple for modeling zeolite electronic structures.3,4
However, even when clusters are constrained to mimic the
target zeolite, qualitative errors can arise.15-23 We found this
in our study of the O(1) to O(4) jump in H-Y modeled with
© 2005 American Chemical Society
Published on Web 08/20/2005
Modeling Proton Transfer in Zeolites
J. Chem. Theory Comput., Vol. 1, No. 6, 2005 1233
a “3T” cluster [tSi-O(1)-Al(OH)2-O(4)-Sit].24-26 Although protons are observed by neutron diffraction at O(1)
but not at O(4),27 our O(1) to O(4) reaction energies were
found to be negligible, signaling a qualitative failing of the
small cluster model. In addition to ignoring long-range forces
and some hydrogen bonding, small clusters lack the steric
constraints that characterize adsorption in zeolites, because
the cluster may not represent enough of the zeolite cavity.
Several remedies can be considered for solving the
problems of small cluster models. The simplest is to use
progressively larger clusters in vacuo until desired properties
converge.21,28 It is clear that with increasing cluster size, more
of the important local electronic and steric effects are
included, while the effect of the terminating region diminishes when it is further separated from the proton transfer
site. In that approach, one has the choice of fully relaxing
the clusters or imposing some geometrical constraints based
on structural data.
Two promising avenues for including long-range forces
are periodic quantum calculations29-31 and embedded
clusters.32-34 The periodic approach, which is typically based
on density functional theory (DFT), is relatively straightforward and provides the only reliable method for testing
approximations other than those in DFT itself.35-37 However,
in the present context periodic DFT calculations suffer from
two main drawbacks: they are expensive, treating all atoms
equally, even distant spectator atoms; and they can be applied
only to zeolites with sufficiently small unit cells, wherein
too many acid sites may be formed. These limitations steer
us to the embedded cluster approach.
The embedded cluster idea generally involves breaking a
total system (S) into an inner region (I) of chemical interest
and an outer environment (O). For zeolites and other
covalent-network solids, this partitioning inherently leaves
dangling bonds, requiring special methods to saturate the
bonding in I. In many implementations of embedding,38 link
atoms are added to the inner region, yielding a cluster (C).
One then endeavors to simulate the cluster C with chemical
accuracy (hi), while modeling region O with much cheaper
methods (lo). The total potential energy is then approximated
by
Eembed ) Elo(S) + [Ehi(C) - Elo(C)]
(1)
) Ehi(C) + [Elo(S) - Elo(C)]
(2)
Although eqs 1 and 2 are identical, they offer different
viewpoints on embedding. Equation 1 suggests a low-level
treatment of the total system corrected by a high-level cluster
calculation, while eq 2 implies a high-level cluster calculation
corrected by a low-level treatment of the environment.
Although the embedding approach is appealing, two main
issues about embedding in zeolite science remain generally
unknown: optimal low- and high-levels of theory and
optimal sizes of cluster C and total system S. Below we take
a systematic approach at addressing this latter issue; in a
forthcoming publication, we will report on optimal low- and
high-levels of theory for modeling proton transfer in zeolites.
Sauer and co-workers have published many seminal
calculations of proton-transfer energies in bare zeolites.39
Their recent work on embedding clusters with their QMPot code is based on the following approach:12,13,39,40 (i) use
periodic boundary conditions for the total system; (ii) use a
well-tuned molecular mechanics potential for the low level
of theory, preferably one that has been fitted to electronic
structure data obtained at the same level used for quantum
cluster calculations; (iii) use a shell-model potential for the
low level, to account for some electronic polarization;41,42
(iv) use a reactive potential function for the low level, to
mimic energies associated with making and breaking bonds;43,44
and (v) do not electronically polarize the quantum cluster
by its environment, to avoid double counting such polarization which is already treated approximately at the low level.
Using this approach, Sierka and Sauer calculated reaction
energies and barriers for specific proton jumps in acidic
chabazite, H-Y and H-ZSM-5.13 For the O(1) to O(4) jump
in H-Y zeolite, they obtained reaction energies in the range
10-22 kJ mol-1 and a bare barrier of 95-100 kJ mol-1,
and found that proton transfer barriers are generally increased
by long-range forces.12,13
We have initiated a research program modeling reactions
catalyzed by zeolites of various chemical compositions.35 Our
present study of proton transfer in bare zeolites represents a
base case for calibrating methods for future study. To
promote the widest applicability by zeolite modelers, we seek
to benchmark embedding methods suitable for use with all
elements without further parametrization. This represents a
departure from the philosophy behind methods such as QMPot, which employ reactive force fields finely tuned for
specific systems. This paper is our first in a series using the
ONIOM embedding procedure38 in the Gaussian quantum
chemistry code.45,46 We believe this to be a versatile
combination for modeling zeolites47-49 because the implementation of ONIOM in Gaussian allows partitioning into
more than two layers and provides molecular mechanics,
semiempirical and ab initio methods for each level of theory.
In the present article, we show that reliable energies of proton
transfer in zeolites can be obtained using smaller clusters
embedded in larger clusters and using the universal force
field (UFF),50 a generic nonreactive force field, as the low
level of theory. This represents progress toward a simpler
prescription for using embedding methods to model zeoliteguest systems of arbitrary composition.
We compare results from constrained and embedded
quantum clusters of the same size. In constrained clusters,
terminal atoms are fixed at positions suggested by diffraction
experiments, while embedded clusters are connected to larger
networks. We refer to these as ‘hard’ and ‘soft’ restraints,
respectively. By comparing clusters of the same size with
hard and soft restraints, we determine how various methods
of terminating a quantum cluster influence proton-transfer
properties. One might imagine that imposing hard restraints
would perturb structural and vibrational properties of proton
transfer. To address this issue, we compute proton-transfer
attempt frequencies and O-Al-O angles in most systems
studied.
In the present study we do not include electrostatic
interactions between the quantum region and its environment.
In a forthcoming publication we will include these electro-
1234 J. Chem. Theory Comput., Vol. 1, No. 6, 2005
Fermann et al.
Figure 1. (a) 8T cluster embedded in a 53T system, also denoted 8T-16.95 Å, where 16.95 Å is the radius of the smallest
sphere containing the entire system, with oxygens O(1) and O(4) labeled. (b) 11T cluster embedded in a 53T system
(11T-16.95 Å). (c) 16T cluster embedded in a 53T system (16T-16.95 Å).
statics, incorporating charges from region O in the Hamiltonian of the quantum cluster.51 Studying embedded clusters
in this stepwise fashion allows us to disentangle mechanical
and electronic embedding effects, thereby shedding further
light on the energetics of reactions in zeolites.
The remainder of this article is organized as follows: in
section II, we describe the zeolite clusters, electronic structure
methods, and types of calculations performed. Section III
details the results and discusses their implications, and in
section IV we give concluding remarks.
II. Computational Methods
In this section, we describe the methods used to study the
O(1) to O(4) proton jump in H-Y zeolite. First, we detail
the zeolite clusters investigated and the methods used to
incorporate properties of an extended system. We then review
the electronic structure methods and geometry optimization
techniques employed to compute energies, structures, and
vibrational frequencies.
A. Zeolite Cluster Models. The chemical system under
investigation is H-Y zeolite in the low-Al limit, where a
single Si atom in the entire framework is replaced with an
Al and charge balanced with a proton. The reaction in
question is an internal proton transfer between crystallographically distinct oxygen atoms, O(1) and O(4). All of our
models therefore feature a single Al tetrahedral site surrounded by a siliceous framework of varying size.
We study this internal proton transfer in quantum mechanically modeled clusters ranging in size from 3 tetrahedral
centers (3T ) Al + 2Si) to 22 tetrahedral centers (22T )
Al + 21Si). We construct each cluster by clipping pieces
from a crystal structure27 of H-Y zeolite and terminating
with hydrogen atoms at appropriate distances from atoms
with dangling bonds. In constrained clusters, terminal
hydrogens are placed along the vector toward the next atom
in the zeolite framework at distances of 0.9 and 1.4 Å for
Si-H and SiO-H termination, respectively. If a cluster is
terminated primarily with Si-H bonds, the cluster is denoted
as a “T-H” cluster; the O-H cluster termination is marked
with “T-OH.” Atoms were included in a cluster based on
their distance from the Al. When necessary, atoms were
added to complete a zeolite ring, which avoids placing
terminal atoms in too close proximity when terminating
dangling bonds. Figure 1 shows examples of various cluster
sizes.
Terminating a molecular cluster fails to include the effects
of excluded atoms; we attempt to include the most important
of these effects through the use of geometric constraints
placed on the cluster exterior. As outlined in the Introduction,
we compare the effect of hard and soft restraints to gauge
their influence on reaction energies. (We do not include fully
relaxed clusters in our study, because they relax to rather
nonzeolitic structures, especially for small clusters.) In cluster
calculations with hard restraints (hereafter denoted “constrained”), the terminal hydrogen atoms are frozen in place,
while all other atoms are allowed to relax. As in our previous
work,24 this serves to mimic the covalent “footprint” of the
extended zeolite from which the cluster was clipped. Soft
restraints are enforced by embedding a smaller cluster in a
larger one (hereafter denoted “embedded”), using the Own
N-Layer Integrated Molecular Orbital - Molecular Mechanics (ONIOM) method in GAUSSIAN quantum chemistry
codes.45,46 The embedded clusters differ from those in the
constrained cluster calculations in the placement of the
terminal hydrogens. In the case of constrained clusters, the
terminal hydrogens are fixed in place through the entire
optimization procedure, at positions determined from the
crystal structure. In the case of embedded clusters, terminal
hydrogens, also referred to as link atoms, are placed along
the bond vectors pointing from the last shell of atoms in the
cluster to the first shell of atoms in the outer ONIOM layer.
Link atom positions vary during optimization because the
outer ONIOM layer is itself flexible, except for terminal
atoms on the boundary of the total system, whose positions
are fixed at crystallographic locations.
B. Electronic Structure Methods. In an earlier study, we
reported that constrained cluster models treated at the
B3LYP/6-311G(d,p) level of theory yield accurate results
for geometrical parameters and vibrational frequencies.24 We
also found that convergence of electronic energies is best
obtained through the focal point method by correcting MP2/
6-311G(d,p) energies with the difference {E[MP4/6-31G(d)]
- E[MP2/6-31G(d)]}. In that same publication, we found
that B3LYP/6-311G(d,p) underestimates reaction barriers by
about 10 kJ mol-1 for this zeolite system, which is about
10% of the classical barrier height. This accuracy is deemed
acceptable for the purposes of this study because such
calculations provide a very good compromise between
computational efficiency and accuracy. We thus employed
the B3LYP/6-311G(d,p) level of theory as our high level
throughout this work. Our ONIOM calculations use this basis
set and level of theory to compute Ehi(C), while the Universal
Force Field50 (UFF) level of theory is used to compute Elo(S)
and Elo(C) in eqs 1 and 2.
Modeling Proton Transfer in Zeolites
The choice of UFF for the low level of theory in our
ONIOM calculations is justified in several ways. First, UFF
contains parameters for all elements in the periodic table and
as such provides a broadly applicable low level of theory.
Though we do not exploit this generality in the present study,
it may prove useful in future work. Second, using UFF allows
us to test whether a nonreactive force field can produce
accurate proton-transfer energies within ONIOM. The form
of the ONIOM energy in eq 2 suggests this is possible,
because unphysical energy terms that are introduced by bond
breaking events identically cancel in the correction term
[Elo(S) - Elo(C)]. Third, UFF as parametrized for silicates
lacks partial charges. As such, this low level of theory is
essentially a ball-and-spring model with minimal electronic
effects, making it ideal for determining the influence of only
network constraints, without additional complexities associated with long-range forces. This is consistent with our
overall stepwise investigation into the various effects of the
environment on proton transfer.
C. Calculation Details. We investigated convergence of
four reaction parameters with respect to cluster sizes. These
parameters are the reaction energy, activation energy,
optimized O(1)-Al-O(4) angles for protons at O(1) and at
O(4), and the vibrational frequency most closely associated
with the O(1) to O(4) proton-transfer reaction coordinate.
To compute these parameters, geometry optimizations of
each cluster model were performed, followed by a vibrational
frequency analysis.
The reaction energy and activation energy were computed
as ∆Erxn ) EHO(4) - EHO(1) and ∆Eact ) ETS - EHO(1),
respectively. The reported energies do not include zero-point
or thermal corrections. From these calculations, the O(1)Al-O(4) angles at the reactant and product geometries were
also recorded. In each individual geometry optimization, the
cluster energy was converged to 10-6 Hartrees and the RMS
force to 10-4 Hartrees ao-1. All transition states were
confirmed to be first-order saddle points by computing and
diagonalizing the mass-weighted Hessian matrix. We calculated vibrational frequencies with the masses of terminal
atoms set to 106 au to mimic the macroscopic mass of the
zeolite framework. For a subset of optimized constrained and
embedded clusters, we report vibrational frequencies associated with the early stages of proton transfer, i.e., proton
transfer “attempt frequencies”.
Most embedding or QM/MM packages use a two-step or
double-iteration method for geometry optimization.34,52 In
the first step, a geometry displacement is made only in the
QM region (region I), while the MM region (region O) is
kept frozen. For this step all the usual “small molecule”
geometry optimization techniques are employed, such as a
second-order approximation of the potential surface, Hessian
update mechanisms, and the use of redundant internal
coordinates. We note that QM/MM forces are used to
determine the QM step, not just the QM forces. In the second
step, the MM region is fully optimized, while the QM region
is kept frozen. For this step we use a conjugate gradient
optimizer in Cartesian coordinates, which is suitable for large
systems. The two steps are repeated until convergence is
reached.
J. Chem. Theory Comput., Vol. 1, No. 6, 2005 1235
The main reason for using this double-iteration method is
that it remains intractable to use a second-order optimizer
in redundant internal coordinates for very large systems. The
required inversion or diagonalization of the Hessian is a
computational bottleneck for larger systems, and the coordinate transformations become prohibitively expensive. With
the double-iteration method, however, we still use a very
accurate optimization method for the QM region, keeping
the number of (expensive) QM energy and gradient evaluations to a minimum. The number of MM energy and
gradient evaluations will increase significantly, but since
these are several orders of magnitude cheaper than QM
calculations, they have little effect on the total computational
time.
The method outlined above, which uses relatively few
“macroiterations” for the QM region and several more
“microiterations” for the MM region, has been implemented
in several QM/MM packages and has been used in numerous
QM/MM studies. There are, however, several serious drawbacks of this scheme, related to the fact that the Hessian is
only updated for the QM region. First, large displacements
in the MM region, especially when a different local minimum
is found, can lead to numerical instability of the Hessian
update. Second, in the QM step, there is no direct coupling
with the MM region, which deteriorates the quality of the
QM step and increases the number of steps needed to reach
convergence. Recently, we have addressed these issues by
developing techniques that explicitly include coupling between the two regions in the QM step, while still using the
double-iteration scheme.53 This has significantly improved
convergence behavior; indeed, we find no significant difference in performance between regular QM geometry
optimization and QM/MM geometry optimizations. All the
calculations were performed with GAUSSIAN0345 and
Gaussian Development Version46 on Intel Linux workstations.
III. Results and Discussion
In this section, we describe the optimized geometries,
reaction and activation energies, and vibrational frequencies
obtained by the computational methods described above for
modeling proton transfer from O(1) to O(4) in H-Y zeolite.
We note that both experimental data and previous calculations agree that the reaction energy for this transfer should
be endothermic, by 10-22 kJ mol-1 according to Sauer et
al.,12 which provides a broad target for our convergence
studies.
A. Geometric Parameters. The O(1)-Al-O(4) angle
affects proton-transfer kinetics by determining the jump
distance and hence the activation energy of the reaction.
Tables 1 and 2 show the values of O(1)-Al-O(4) angles
from the constrained and embedded clusters for a variety of
sizes. In Table 1, the O(1) results show reasonable convergence, and only the smallest (5T-H and 8T-H) constrained
clusters are slightly off. In contrast, the angles of the
embedded clusters (Table 2) are constant throughout the
series, including the smallest clusters.
B. Reaction Energies. Table 3 summarizes our results
for the proton-transfer reaction energy for both the con-
1236 J. Chem. Theory Comput., Vol. 1, No. 6, 2005
Fermann et al.
Table 1. OAlO Angle at Optimized Geometry from
Constrained Cluster
fashion as seen in the constrained cluster calculations. Here,
the values for each total system size approach convergence
at the 8T-OH quantum cluster size. Again, we observe that
describing larger sections of the total system with a quantum
mechanical method does not markedly improve the final
result.
In our work all the larger clusters are terminated with
hydroxyl groups; only for 8T can we compare the effect of
hydrogen termination vs hydroxyl termination. We see indeed
a large difference in the embedded 8T data, which was also
reported by Brand et al. for proton affinities of acid sites in
ZSM-5.54
In general, we find that a finite cluster, whether constrained
or embedded, appears to be sufficient for predicting proton
transfer reaction energies in zeolites. The use of methods
based on periodic boundary conditions actually masks the
length scales beyond which interactions no longer influence
proton transfer. Determining such length scales is of fundamental importance for understanding how much of the zeolite
actually controls reaction dynamics. The concept of a finite
interaction length may come as a surprise considering that
many researchers employ Ewald sums when simulating
Coulombic interactions in partially ionic media such as
zeolites.55 Such long-range summations are crucial for
calculating absolute adsorption energies of molecules in
zeolites relative to vacuum. However, for calculating energy
differences between nearby configurations of an adsorbed
species, for example a reaction energy or barrier, the present
results suggest that long-range interactions essentially cancel,
leaving an energy difference controlled by local electronic
interactions.
Our data suggest that this proton-transfer reaction is
sensitive to atoms within ca. 5 Å of the reaction center. The
presence of guest molecules will likely change this cutoff
distance, but even in this more complex scenario we still
expect a finite cluster to capture the reaction energetics. It
is reasonable to surmise that using a low level of theory that
includes electronic interactions would enable the use of even
smaller inner regions in embedded calculations. We explore
this possibility in future work.
C. Activation Energies. We calculated activation energies
for constrained clusters and a subset of embedded clusters
(Table 4). As discussed above, all transition states were
confirmed to be first-order saddle points by the usual Hessian
analysis. For the embedded clusters we considered total sizes
of 53T and 166T. Although the number of data points is
5T-H
8T-H
8T-OH
11T-OH
16T-OH
22T-OH
H on O(1)
H on O(4)
93
85
105
98
99
102
93
86
98
96
100
103
Table 2. OAlO Angle at Optimized Geometry from
Embedded Cluster
H on O(1)
H on O(4)
99
101
100
100
101
99
105
98
97
100
5T(H)-53T
8T(OH)-53T
16T(OH)-53T
16T(OH)-259T
22T(OH)-259T
strained and embedded clusters. Reaction energies of constrained zeolite clusters show stable convergence, reaching
a value of 14 ( 2 kJ mol-1 at the 11T cluster size. The
anomalously large reaction energy found for both 8T-H and
8T-OH clusters is due to a significant distortion of the
reactant optimized structure allowed by floppiness of dihedral
angles in terminal OSiOH groups. This effect is lessened
both in larger clusters and in the ONIOM calculations. All
constrained calculations correctly show that the transfer is
endothermic. Extending the constrained cluster to include
22T atoms does not significantly affect ∆Erxn, suggesting
that convergence with respect to cluster size is reached. This
suggestion is further corroborated by the ONIOM results
below.
Using the ONIOM method, we seek convergence with
respect to both the quantum cluster size and the total system
size, which are represented as the rows and columns of Table
3, respectively. By scanning across any row, it is clear that
changing the total system size has little effect on the reaction
energy. At first this may be surprising, because the larger
calculations include an order of magnitude more atoms.
However, because the lower level of theory employed is UFF
and therefore lacks long range forces in our calculations,
there is only a local mechanical coupling between the inner
and outer layers of the ONIOM model.
The size of the inner layer does, however, strongly affect
the calculated reaction energy in approximately the same
Table 3. ∆Erxn for O(1) to O(4) Proton Transfera
embedded (with size of full system in parentheses)
cluster size
constrained
3T-H
5T-H
8T-H
8T-OH
11T-OH
16T-OH
22T-OH
2.5
4.3
22.6
26.7
13.6
14.6
13.8
a
In kJ mol-1.
23 T
(7 Å)
53 T
(10 Å)
98 T
(12.44 Å)
166T
(14.8 Å)
259T
(16.95 Å)
439T
(19.91 Å)
1.3
5.4
8.4
1.3
5.9
9.9
14.7
14.4
2.0
6.9
10.7
15.4
14.9
1.8
6.5
10.3
15.4
15.0
1.8
6.4
10.7
15.2
14.8
18.3
15.4
1.6
6.3
10.6
Modeling Proton Transfer in Zeolites
J. Chem. Theory Comput., Vol. 1, No. 6, 2005 1237
Table 4. ∆Eact for O(1) to O(4) Proton Transfera
embedded (with total size)
cluster size
constrained
3T-H
5T-H
8T-H
8T-OH
11T-OH
16T-OH
22T-OH
90.2
93.6
57.3
138.8
133.4
131.6
138.8
a
53T (10 Å)
128.3
137.8
133.7
Table 5. Vibrational Frequency in Wavenumbers (cm-1)
for Constrained Cluster
H on O(1)
166T (14.8 Å)
118.9
126.8
134.7
139.2
136.5
132.7
In kJ mol-1.
smaller than for the reaction energy, this still allows us to
investigate convergence for both the total and quantum
cluster size. From Table 4 we see that for embedded clusters,
independent of total cluster size, the barrier converges at
quantum cluster size 8T-OH. This was also the case for the
reaction energy treated by embedded clusters (Table 3). The
constrained barrier calculations also converge at quantum
cluster size 8T-OH. These results further suggest that finite
clusters, either constrained or embedded, can produce stable
energetics of proton transfer in zeolites.
Our converged barrier is about 135 kJ mol-1 for both
embedded and constrained clusters. For comparison, Sierka
and Sauer obtained a barrier of about 100 kJ mol-1 for the
same proton jump.13 About 10 kJ mol-1 of this difference is
a basis set effect. This was determined by repeating our
computation of the barrier height in the 8T(OH)-53T
embedded cluster using the Alrichs basis set used by Sierka
and Sauer. This gave a barrier of 127.0 kJ mol-1, as
compared to 137.8 kJ mol-1 obtained using the 6-311G(d,p)
basis set (see Table 4).
The remaining difference in barriers is harder to pin down.
Our calculations and those of Sierka and Sauer differ mainly
in the way volume is constrained and in the treatment of
outer layer atoms. Regarding the former, we have built
embedded and constrained zeolite clusters from diffraction
data, fixing terminal atoms at experimentally determined
locations. In contrast, Sierka and Sauer apply periodic
boundary conditions at constant volume, with the lattice
parameter determined from an initial force field optimization
of the H-Y system at constant pressure. It is possible that
our method of fixing terminal atoms can impose strain on
reactant and transition state configurations. Such strain is
expected to diminish as the total system size increases, thus
placing terminal atom constraints farther away from the
reaction center. However, the fact that barriers from our 8TOH and 11T-OH quantum clusters (constrained and embedded) remain stable with respect to total system size suggests
that our method of constraining volume does not impose
unphysical strain on this proton-transfer process.
Our calculations and those of Sauer and Sierka also differ
in the treatment of outer layer atoms. Our calculations
represent electronic effects only in the quantum cluster, while
those of Sierka and Sauer include classical electrostatics in
the outer layer as well. It is possible that our local treatment
of electronic effects can introduce errors into reactant and
transition state energies. Such errors are expected to diminish
as the quantum cluster size increases. The fact that barriers
5T-H
8T-OH
11T-OH
1054
1039
1011
Table 6. Vibrational Frequency in Wavenumbers (cm-1)
for Embedded Cluster
5T(H)-23T
8T(H)-53T
H on O(1)
H on O(4)
1064
1037
1041
1030
Table 7. Calculation Times for Optimizing the System
with the Proton at O(1)
system
av time per
optimization
cycle (h)
no. of
optimization
cycles
total
CPU
time (h)
22T-OH constrained
22T(OH)-259T embedded
11T-OH constrained
11T(OH)-259T embedded
8T-OH constrained
8T(OH)-53T embedded
10.5
10.7
4.5
4.9
2.5
2.6
60
20
35
20
60
20
630
213
155
98.8
153
51.8
from quantum clusters 8T-OH and larger (constrained and
embedded in Table 4) remain stable with respect to quantum
cluster size suggests that our calculations include the
electronic effects relevant for this proton transfer process.
To pursue this point further, we will report in a forthcoming
publication the results of fully periodic quantum calculations
on this system.15
D. Vibrational Frequencies. Vibrational frequencies are
important dynamical parameters for quantifying activation
entropies and attempt frequencies. Vibrational frequency
analyses were performed on selected constrained and embedded clusters to test the convergence behavior of frequencies
with respect to cluster size. Tables 5 and 6 show the
vibrational frequency of the normal mode with the largest
component of Al-O-H wag for the proton situated at O(1),
which corresponds closely with the proton-transfer reaction
coordinate.
Both sets of calculations show very similar vibrational
frequencies for the H wag. The variation from one cluster
size to another is smaller than the expected uncertainty of
the electronic structure method. This is a bit surprising,
considering that the hard termination inherent in constrained
clusters might be expected to shift these to higher frequencies. However, our results suggest that either method of
applying geometric constraints to the clusters is sufficient
to achieve convergence of this particular dynamical parameter.
E. Computational Time Comparisons. In Table 7, we
show CPU time comparisons for pairs of constrained and
embedded calculations that contain identical quantum clusters. The times presented are for geometry optimizations
starting from the crystal structure, with the acidic hydrogen
added in a reasonable position relative to O(1). All timing
1238 J. Chem. Theory Comput., Vol. 1, No. 6, 2005
calculations were performed on dual processor 2.8 GHz
PIV computers with 1 GB of core memory running Linux
RedHat 9. Embedded cluster timings were obtained with the
quadratically coupled QM/MM geometry optimization
method.45,46,53 For purposes of comparing CPU times, the
optimizations were considered converged when the energy
is stable to 0.00005 H (0.13 kJ mol-1), which is the degree
of precision we seek in our reaction energies. We allowed
the optimizations to run significantly longer to test the
robustness of this criterion and found only negligible changes
in structure or energy. In Table 7, we also show the number
of macroiterations required for geometry optimizations and
the average CPU time required for each macroiteration.
In principle, we expect very similar times per optimization
cycle for identical quantum clusters regardless of the method
of constraint, because in the embedded calculations, the time
used to compute energies and forces using UFF is negligible
compared to the time spent in the quantum part of the
calculation. This is precisely what we observe in Table 7.
These timing data suggest that the principal difference
between a constrained calculation and an embedded calculation on identical quantum clusters is simply the number of
optimization cycles required to achieve convergence. In this
respect, the constrained calculations appear to be much
slower since they require many more optimization cycles.
This poor convergence behavior may arise from a mismatch
between floppy OSiOH dihedral angles and fixed terminal
hydrogen atoms. More work is required to better understand
this phenomenon.
Perhaps the most straightforward and important comparison in Table 7 is between the constrained and embedded
cluster calculations that give converged reaction energies in
the least time. These are the constrained 11T-OH and
embedded 8T(OH)-53T systems, which converge in 155 and
51.8 CPU h, respectively. Thus, embedding speeds up these
geometry optimizations by a factor of 3, despite the use of
a generic low-level of theory (UFF) in ONIOM calculations.
IV. Concluding Remarks
We have studied the convergence properties of embedded
and constrained cluster models of proton transfer in zeolites.
We applied density functional theory to describe clusters and
ONIOM to perform the embedding. We focused on converging the reaction energy and barrier of the O(1) to O(4) jump
in H-Y zeolite as well as vibrational and structural aspects
of this jump. We found that using successively larger clusters
in vacuo gives convergence of the reaction energy to 14 (
2 kJ mol-1, and the barrier to 135 ( 5 kJ mol-1, as long as
the clusters are constrained to mimic zeolitic structures. These
calculations converged for clusters with radii larger than 5
Å, containing at least 11 tetrahedral (Si or Al) atoms. We
embedded quantum clusters of various sizes in larger clusters
with total radii in the range 7-20 Å, using the universal
force field (UFF) as the lower level of theory in ONIOM.
We found convergence of the proton-transfer energy without
the use of reactive force fields or periodic boundary
conditions in the embedding procedure. Embedded cluster
calculations gave converged reaction energies for quantum
clusters containing at least 8 tetrahedral atoms. Optimizations
Fermann et al.
on embedded clusters required many fewer cycles, and hence
much less CPU time, than did optimizations on comparable
constrained clusters.
Our present results suggest a reaction energy of 14 ( 2
kJ mol-1 and a barrier of 135 ( 5 kJ mol-1 for the O(1) to
O(4) proton jump in H-Y zeolite. The smallest systems that
yield reaction energies converged to within 1 kJ mol-1 of
this value are the 11T-OH constrained cluster and the
8T(OH)-53T embedded cluster. These quantum clusters
include atoms within ca. 5 Å of the reactive center. In future
work, we will explore whether greater computational efficiency can be obtained by splitting the reactive region into
two layers described by an accurate ab initio theory and a
cheaper electronic theory, thereby giving a three-layer
ONIOM calculation.56 In the end, we aim for a simple and
user-friendly method for modeling a wide variety of reactions
in zeolites.
Acknowledgment. S.M.A. thanks the UMass Amherst
Department of Chemistry for generous funding of our
computer laboratory.
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